Prove that a closed equipotential surface

Question:

Prove that a closed equipotential surface with no charge within itself must enclose an equipotential volume.

Solution:

In a closed equipotential surface, the potential changes from position to position.

The potential inside the surface is different from the potential gradient caused in the surface that is dV/dr

This also means that electric field is not equal to zero and it is given as E = -dV/dr

Therefore, it could be said that the field lines are either pointing inwards or outwards the surface.

So, it can be said that the field lines originate from the charges inside which contradicts the original assumption. Therefore, the volume inside the surface must be equipotential.

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